Problem 88 - College Algebra

Question

For Exercises \(86-89\) use your graphing utility to graph. each side of the equations in the same viewing rectangle. Based on the resulting graph, label each equation as conditional, inconsistent, or an identity. If the equation is conditional, use the \(x\) -coordinate of the intersection point to find the solution set. Verify this value by direct substitution into the equation. $$ 2\left(x+\frac{1}{2}\right)=5 x+1-3 x $$

Step-by-Step Solution

Answer

The equation is an identity. The solution is all real numbers.

Step 1: Simplifying the Equation

Firstly, distribute the 2 in the left hand side (LHS) of the equation to obtain \(2x + 1\). Then, simplify the right hand side (RHS) to get \(2x + 1\). Thus, the equation simplifies to \(2x + 1 = 2x + 1\).

Step 2: Graphing the Two Sides

When we graph \(y = 2x + 1\) (representing both the LHS and RHS of the equation), it is found that it is a single straight line. This means the equation is an identity since the two sides coincide.

Step 3: Interpretation

As the two sides are exactly the same, every x-value is a solution to the equation, since for every x the LHS and RHS will be equal. So the solution is all real numbers.

Step 4: Verification

We can verify this by substituting any random real number for x into the equation. For example, let's use \(x = 0\). Substituting into the equation results in \(2*0 + 1 = 1\) and \(2*0 + 1 = 1\), which confirms that any real number can be a solution.

Key Concepts

Understanding Graphing Utilities

A graphing utility is an essential tool in mathematics, especially for understanding the behavior and nature of equations. It allows you to effectively visualize equations by plotting their graphs. By entering the equation into a graphing utility, it translates the algebraic equation into a visual format, aiding comprehension.
Here are some key points about using graphing utilities:

• Graphing utilities can be physical calculators or online programs.
• They help to visually compare two different equations or sides of an equation.
• You can identify where two graphs intersect, which is crucial for determining solutions to equations.

In our exercise, we used a graphing utility to plot both sides of the equation. This helps us see that they form the same line, which leads us to identify what kind of equation we're dealing with.

Exploring Conditional Equations

A conditional equation is one that is true only for certain values of the variable. This means that it's not universally valid but instead relies on specific instances to hold true. For instance, if you have an equation where both sides are equal only when \(x\) equals a particular number, then it's conditional.
Understanding these points can help you identify a conditional equation:

• The graphs intersect at specific points, not along the whole line.
• The equation does not simplify to an identity or inconsistency.
• Conditional equations have a finite number of solutions.

In general, finding the intersection point between the graphs' lines will give you the solution set for conditional equations.

Recognizing Inconsistent Equations

An inconsistent equation is one that has no solution at all. When you graph such an equation, the lines representing the equation do not intersect. This lack of intersection indicates there are no values for \(x\) that satisfy both sides of the equation simultaneously.
Remember these traits to spot an inconsistent equation:

• The graphs of the equations are parallel, meaning they'll never meet.
• Upon simplifying, the equation results in a false statement like \(0 = 5\).
• No value of \(x\) will make both sides equal.

In the case of the given exercise, had the equation been inconsistent, the graphs could not have overlapped or intersected at any point.

Defining Identity Equations

An identity equation stands out because it is true for all values of the variable involved. It means no matter what \(x\) you substitute, both sides of the equation will always be equal. This was the case in our exercise. Here, both sides simplified to the same expression, \(2x + 1\).
Let's identify the key features of identity equations:

• The graph of each side overlaps entirely.
• Any value for the variable will satisfy the equation.
• It simplifies to a tautological truth like \(2x + 1 = 2x + 1\).

Thus, with identity equations, every real number is part of the solution set, making them universally true across all contexts.